Transfer Functions of One-Dimensional Distributed Parameter Systems

1992 ◽  
Vol 59 (4) ◽  
pp. 1009-1014 ◽  
Author(s):  
B. Yang ◽  
C. A. Tan
Keyword(s):  
Transfer Function ◽  
Transfer Functions ◽  
Distributed Parameter Systems ◽  
Axially Moving Beam ◽  
System Response ◽  
Distributed Parameter ◽  
One Dimensional ◽  
Adjoint Systems ◽  
The Laplace Transform ◽  
Axially Moving

Distributed parameter systems describe many important physical processes. The transfer function of a distributed parameter system contains all information required to predict the system spectrum, the system response under any initial and external disturbances, and the stability of the system response. This paper presents a new method for evaluating transfer functions for a class of one-dimensional distributed parameter systems. The system equations are cast into a matrix form in the Laplace transform domain. Through determination of a fundamental matrix, the system transfer function is precisely evaluated in closed form. The method proposed is valid for both self-adjoint and non-self-adjoint systems, and is extremely convenient in computer coding. The method is applied to a damped, axially moving beam with different boundary conditions.

Transfer Function Modeling of Distributed Piezoelectric Vibration Energy Harvesters

2009 ◽  
Author(s):  
Chin An Tan ◽  
Heather L. Lai
Keyword(s):  
Transfer Function ◽  
Transfer Functions ◽  
Vibration Energy ◽  
System Response ◽  
Distributed Parameter ◽  
Domain Modeling ◽  
Vibration Energy Harvesting ◽  
Energy Harvesters ◽  
Adjoint Systems ◽  
The Stability

Extensive research has been conducted on vibration energy harvesting utilizing a distributed piezoelectric beam structure. A fundamental issue in the design of these harvesters is the understanding of the response of the beam to arbitrary external excitations (boundary excitations in most models). The modal analysis method has been the primary tool for evaluating the system response. However, a change in the model boundary conditions requires a reevaluation of the eigenfunctions in the series and information of higher-order dynamics may be lost in the truncation. In this paper, a frequency domain modeling approach based in the system transfer functions is proposed. The transfer function of a distributed parameter system contains all of the information required to predict the system spectrum, the system response under any initial and external disturbances, and the stability of the system response. The methodology proposed in this paper is valid for both self-adjoint and non-self-adjoint systems, and is useful for numerical computer coding and energy harvester design investigations. Examples will be discussed to demonstrate the effectiveness of this approach for designs of vibration energy harvesters.

Transfer Functions of One-Dimensional Distributed Parameter Systems by Wave Approach

2006 ◽  
Vol 129 (2) ◽  
pp. 193-201 ◽  
Author(s):  
B. Kang
Keyword(s):  
Transfer Function ◽  
Frequency Response ◽  
Wave Reflection ◽  
Transfer Functions ◽  
Distributed Parameter Systems ◽  
Distributed Parameter ◽  
Complex Frequency ◽  
Reflection And Transmission ◽  
One Dimensional ◽  
Analysis Technique

An alternative analysis technique, which does not require eigensolutions as a priori, for the dynamic response solutions, in terms of the transfer function, of one-dimensional distributed parameter systems with arbitrary supporting conditions, is presented. The technique is based on the fact that the dynamic displacement of any point in a waveguide can be determined by superimposing the amplitudes of the wave components traveling along the waveguide, where the wave numbers of the constituent waves are defined in the Laplace domain instead of the frequency domain. The spatial amplitude variations of individual waves are represented by the field transfer matrix and the distortions of the wave amplitudes at point discontinuities due to constraints or boundaries are described by the wave reflection and transmission matrices. Combining these matrices in a progressive manner along the waveguide using the concepts of generalized wave reflection and transmission matrices leads to the exact transfer function of a complex distributed parameter system subjected to an externally applied force. The transient response solution can be obtained through the Laplace inversion using the fixed Talbot method. The exact frequency response solution, which includes infinite normal modes of the system, can be obtained in terms of the complex frequency response function from the system’s transfer function. This wave-based analysis technique is applicable to any one-dimensional viscoelastic structure (strings, axial rods, torsional bar, and beams), in particular systems with multiple point discontinuities such as viscoelastic supports, attached mass, and geometric/material property changes. In this paper, the proposed approach is applied to the flexural vibration analysis of a classical Euler–Bernoulli beam with multiple spans to demonstrate its systematic and recursive formulation technique.

Transfer Function Analysis of Non-Uniformly Distributed Parameter Systems

1993 ◽  
Author(s):  
Bingen Yang ◽  
Houfei Fang
Keyword(s):  
Distributed Systems ◽  
Transfer Function ◽  
State Space ◽  
Fundamental Matrix ◽  
Function Analysis ◽  
Distributed Parameter Systems ◽  
System Response ◽  
Distributed Parameter ◽  
Transfer Function Analysis ◽  
Function Formulation

Abstract This paper studies a transfer function formulation for general one-dimensional, non-uniformly distributed systems subject to arbitrary boundary conditions and external disturbances. The purpose is to provide an useful alternative for modeling and analysis of distributed parameter systems. In the development, the system equations of the non-uniform system are cast into a state space form in the Laplace transform domain. The system response and distributed transfer functions are derived in term of the fundamental matrix of the state space equation. Two approximate methods for evaluating the fundamental matrix are proposed. With the transfer function formulation, various dynamics and control problems for the non-uniformly distributed system can be conveniently addressed. The transfer function analysis is also applied to constrained/combined non-uniformly distributed systems.

scholarly journals A general transfer function representation for a class of hyperbolic distributed parameter systems

2013 ◽  
Vol 23 (2) ◽  
pp. 291-307 ◽  
Author(s):  
Krzysztof Bartecki
Keyword(s):  
Transfer Function ◽  
Transfer Functions ◽  
Function Analysis ◽  
Semigroup Theory ◽  
Distributed Parameter Systems ◽  
Hyperbolic Partial Differential Equations ◽  
Distributed Parameter ◽  
Function Representation ◽  
Tube Heat Exchanger ◽  
Transfer Function Analysis

Results of transfer function analysis for a class of distributed parameter systems described by dissipative hyperbolic partial differential equations defined on a one-dimensional spatial domain are presented. For the case of two boundary inputs, the closed-form expressions for the individual elements of the 2×2 transfer function matrix are derived both in the exponential and in the hyperbolic form, based on the decoupled canonical representation of the system. Some important properties of the transfer functions considered are pointed out based on the existing results of semigroup theory. The influence of the location of the boundary inputs on the transfer function representation is demonstrated. The pole-zero as well as frequency response analyses are also performed. The discussion is illustrated with a practical example of a shell and tube heat exchanger operating in parallel- and countercurrent-flow modes.

scholarly journals Dynamic control of maximal ventricular elastance via the baroreflex and force-frequency relation in awake dogs before and after pacing-induced heart failure

2010 ◽  
Vol 299 (1) ◽  
pp. H62-H69 ◽  
Author(s):  
Xiaoxiao Chen ◽  
Javier A. Sala-Mercado ◽  
Robert L. Hammond ◽  
Masashi Ichinose ◽  
Soroor Soltani ◽  
...  
Keyword(s):  
Heart Failure ◽  
Transfer Function ◽  
Transfer Functions ◽  
Dynamic Properties ◽  
System Response ◽  
Force Frequency ◽  
Arterial Baroreflex ◽  
Ventricular Elastance ◽  
Arterial Blood ◽  
Frequency Relation

We investigated to what extent maximal ventricular elastance ( Emax) is dynamically controlled by the arterial baroreflex and force-frequency relation in conscious dogs and to what extent these mechanisms are attenuated after the induction of heart failure (HF). We mathematically analyzed spontaneous beat-to-beat hemodynamic variability. First, we estimated Emax for each beat during a baseline period using the ventricular unstressed volume determined with the traditional multiple beat method during vena cava occlusion. We then jointly identified the transfer functions (system gain value and time delay per frequency) relating beat-to-beat fluctuations in arterial blood pressure (ABP) to Emax (ABP→ Emax) and beat-to-beat fluctuations in heart rate (HR) to Emax (HR→ Emax) to characterize the dynamic properties of the arterial baroreflex and force-frequency relation, respectively. During the control condition, the ABP→ Emax transfer function revealed that ABP perturbations caused opposite direction Emax changes with a gain value of −0.023 ± 0.012 ml−1, whereas the HR→ Emax transfer function indicated that HR alterations caused same direction Emax changes with a gain value of 0.013 ± 0.005 mmHg·ml−1·(beats/min)−1. Both transfer functions behaved as low-pass filters. However, the ABP→ Emax transfer function was more sluggish than the HR→ Emax transfer function with overall time constants (indicator of full system response time to a sudden input change) of 11.2 ± 2.8 and 1.7 ± 0.5 s ( P < 0.05), respectively. During the HF condition, the ABP→ Emax and HR→ Emax transfer functions were markedly depressed with gain values reduced to −0.0002 ± 0.007 ml−1 and −0.001 ± 0.004 mmHg·ml−1·(beats/min)−1 ( P < 0.1). Emax is rapidly and significantly controlled at rest, but this modulation is virtually abolished in HF.

Transfer Function Formulation of Constrained Distributed Parameter Systems, Part II: Applications

1993 ◽  
Vol 60 (4) ◽  
pp. 1012-1019 ◽  
Author(s):  
C. H. Chung ◽  
C. A. Tan
Keyword(s):  
Transfer Function ◽  
Elastic Foundation ◽  
Normal Modes ◽  
Distributed Parameter Systems ◽  
Distributed Parameter ◽  
Displacement Method ◽  
Mode Localization ◽  
Function Formulation ◽  
Free Beam ◽  
Symmetric Systems

In this paper, the application of the transfer function formulation and the generalized displacement method (GDM) to the analysis of constrained distributed parameter systems is illustrated. Two kinds of classical examples are considered. In the constrained free-free beam example, it is shown how the GDM gives the eigensolutions without requiring knowledge of the normal modes of the unconstrained beam. In the string on a partial elastic foundation example, mode localization and eigenvalue loci veering phenomena are examined. It is shown that mode localizaation can occur in spatially symmetric systems and for modes whose frequency loci do not veer.

A Transfer-Function Formulation For Nonuniformly Distributed Parameter Systems

1994 ◽  
Vol 116 (4) ◽  
pp. 426-432 ◽  
Author(s):  
B. Yang ◽  
H. Fang
Keyword(s):  
Transfer Function ◽  
State Space ◽  
Fundamental Matrix ◽  
Transfer Functions ◽  
Function Analysis ◽  
System Response ◽  
Approximate Methods ◽  
Arbitrary Boundary ◽  
Transfer Function Analysis ◽  
Function Formulation

This paper studies a transfer-function formulation for general one-dimensional, nonuniformly distributed systems, subject to arbitrary boundary conditions and external disturbances. In the development, the governing equations of the nonuniform system are cast into a state-space form in the Laplace transform domain. The system response and distributed transfer functions are derived in term of the fundamental matrix of the state-space equation. Two approximate methods, the step-function approximation and truncated Taylor series, are proposed to evaluate the fundamental matrix. With the transfer-function formulation, various dynamics and control problems for the nonuniformly distributed system can be conveniently addressed. The transfer-function analysis also is applied to constrained/combined nonuniformly distibuted systems. The method developed is illustrated on two nonuniform beams.

Dynamic Response of an Axially Moving Beam Coupled to Hydrodynamic Bearings

1991 ◽  
Author(s):  
C. A. Tan ◽  
B. Yang ◽  
C. D. Mote
Keyword(s):  
Transfer Function ◽  
Coupled System ◽  
Linear Constraint ◽  
Coupled Systems ◽  
Design Parameters ◽  
Axially Moving Beam ◽  
Constraint Forces ◽  
Hydrodynamic Bearing ◽  
Moving Beam ◽  
Axially Moving

Abstract The vibration response of a distributed axially moving beam, controlled through distributed hydrodynamic bearing forces, is analyzed by the transfer function method. In the axially moving beam, all modes are affected by the distributed bearing coupling. The bearing force is described in terms of impedance functions. An approximate closed-form transfer function for the coupled beam-bearing system is derived. The transfer function is applicable to all linear, one-dimensional, coupled systems, and it is exact for localized, linear constraint forces. The derivation requires knowledge of the transfer function of the axially moving beam which is not available in the literature. A method for determining the beam transfer function is also presented. The frequency response of the coupled system is illustrated for several beam and bearing design parameters.

Distributed Transfer Function Analysis of Complex Distributed Parameter Systems

1994 ◽  
Vol 61 (1) ◽  
pp. 84-92 ◽  
Author(s):  
B. Yang
Keyword(s):  
Closed Form ◽  
Transfer Functions ◽  
Distributed Parameter Systems ◽  
Force Balance ◽  
Distributed Parameter ◽  
Distributed Parameter System ◽  
Parameter System ◽  
Lumped Parameter ◽  
Space Technique ◽  
Transfer Function Analysis

This paper presents a new analytical and numerical method for modeling and synthesis of complex distributed parameter systems that are multiple continua combined with lumped parameter systems. In the analysis, the complex distributed parameter system is first divided into a number of subsystems; the distributed transfer functions of each subsystem are determined in exact and closed form by a state space technique. The complex distributed parameter system is then assembled by imposing displacement compatibility and force balance at the nodes where the subsystems are interconnected. With the distributed transfer functions and the transfer functions of the constraints and lumped parameter systems, exact, closed-form formulation is obtained for various dynamics and vibration problems. The method does not require a knowledge of system eigensolutions, and is valid for non-self-adjoint systems with inhomogeneous boundary conditions. In addition, the proposed method is convenient in computer coding and suitable for computerized symbolic manipulation.

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